Question

The determinant $\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$ if

• A. $x,y,z$ are in A.P.
• B. $x,y,z$ are in H.P.
• C. $x,y,z$ are in G.P.
• D. $xy,yz,zx$ are in A.P.

Answer 1

The correct option is C $x,y,z$ are in G.P.

$\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$
$C_1\to C_1-(p{C}_2+C_3)$
$\left|\begin{array}{ccc}0& x& y\\ 0& y& z\\ -(p^{2}x+2py+z)& xp+y& yp+z\end{array}\right|=0$
$\Rightarrow -(p^{2}x+2py+z)(xz-y^2)=0$
$\Rightarrow xz-y^2=0$
$\Rightarrow xz=y^2$

Hence, $x,y,z$ are in G.P.